Aplicaciones De Integrales

ENGLISH ACTIVITY
APPLICATIONS OF THE DEFINITE INTEGRAL

INTRODUCTION:
The definite integral of a function represents the area bounded by the graph of the function, with a positive sign when the function takes positive values and negative when negative values.


APPLICATIONS:

Arc length:

We will determine the length s of the arc of a curve with equation y = f (x), between points A (f(a)), B (b, f (b)).


As shown in the previous figure, the arc AB divided into n parts, then joining successive points of division by rectilinear segments.


For example, the segment of length would



Then we will have a length aproximaxion the curve AB, by adding:



If we increase indefinitely the number of points of division, then the lengths of the segments tend to zero, sothat:



gives the arc AB, provided that the boundary exists.

To express the limit as an integral we have the following: Suppose that the function with equation y = f (x) is continuous and has continuous derivative at each point of the curve, where A (a, f (a)) to B (b f (b)). Then, by the mean value theorem for derivatives, there is a point D*(x*,y*) between points D and E of the curve,where the tangent is parallel to the string DE, that is:



Then



Which by definition corresponds to the integral:


(we have expressed f '(x) and dy / dx).

Since the length of a curve does not depend on the choice of coordinate axes, if x can be expressed as a function of y, then the arc length is given by:



In each case, calculate the length of arc of a curve that isindicated.



Areas under a cruve:

Let f be a function whose domain is in the closed interval [a, b] such that f(x) ≥ 0 for x ∈ [a, b].
Let R be the plane region bounded by the graphs of the equations: y = f(x), y = 0 (axis x), x = a, x = b.



Let PN be a partition of [a, b] into n subintervals determined by the set {x0, x1, x2, … , xn-1, xn}, with Δ xi = xi - xi-1 , i ∈ {1, 2, … ,n}.

Let Tn = {t1, t2, … , tn} increased Pn.

We construct n rectangles whose bases are the n intervals of the partition Pn whose heights are

f(t1), f(t2), … , f(ti), … , f(tn-1), f(tn)




The area of rectangle i-esimo is given by f(ti) · Δ xi ; and the sum



of the areas of the n rectangles be an approximation to the area A.

If we increase the number of subintervals, thendecreases the length of each subinterval of the partition Pn, obtaining a new addition that will give a greater approximation to the area of R.

Let us now take the following definition:

If f (x) ≥ 0 for x ∈ [a, b] and if a number A such that given a ε > 0, there exists δ > 0 such that


for every partition Pn of [a, b], and any increase in Pn where Np < δ, then this number A is thearea of the region bounded by the graphs of the equation y = f (x), y = 0, x = a, x = b.
Note that this definition must be



and if A exists, then:




Area between two curves:

F and g are functions with domain in the interval [a, b] such that f(x) ≥ g(x) for x ∈ [a, b].
We will determine which is the area of the region R bounded by the graphs of y = f (x), y = g (x), x = a, x =b shown below:



Construct a set of rectangles such that the sum of their areas is an approximation to the area of R.




Let Pn be a partition of [a, b] into n subintervals determined by the set

{x0, x1, x2, …, xi-1, xi, … ,xn-1, xn}

Where Δ xi = xi - xi-1 , i ∈ [1, 2, … ,n].

Let Tn = {t1, t2, … , ti-1, ti , … , tn-1 ,tn} increased Pn. We construct n rectangles whosewidths are the n subintervals of the partition Pn and whose heights are:

f(t1) - g(t1), f(t2) - g(t2), … , f(ti) - g(ti), … , f(tn) - g(tn)

The area of the i-esimo rectangle is [f(ti) - g(ti)] · Δ xi , and the sum of approximation to the area A is given by:



If we increase the number of subintervals, then decreases the length of each subinterval of the partition Pn, obtaining a new...